On a model of an elastic body fully immersed in a viscous incompressible fluid with small data

TL;DR

This paper models an elastic body in a viscous incompressible fluid, proving global well-posedness and exponential decay for small initial data, with the final state being trivial except for a small horizontal displacement.

Contribution

It establishes global existence, uniqueness, and decay estimates for a coupled elastic-fluid system with no stabilization terms, under small initial data.

Findings

Global-in-time well-posedness for small data

Exponential decay to a trivial final state

Final state may include a small horizontal displacement

Abstract

We consider a model of an elastic body immersed between two layers of incompressible viscous fluid. The elastic displacement $w$ is governed by the damped wave equation $w_{tt} + \alpha w_t + \Delta w =0$ without any stabilization terms, where $\alpha >0$, and the fluid is modeled by the Navier-Stokes equations. We assume continuity of the displacement and the stresses across the moving interfaces and homogeneous Dirichlet boundary conditions on the outer fluid boundaries. We establish a~priori estimates that provide the global-in-time well-posedness and exponential decay to a final state of the system for small initial data. We prove that the final state must be trivial, except for a possible small displacement of the elastic structure in the horizontal direction.